How to Explain Functions to Eighth Graders

"Put in 5, get out 11. Put in 3, get out 7. Put in 10, get out 21."

What's the pattern? It's a function—a mathematical machine that transforms inputs into outputs according to a rule. Functions are everywhere in math, science, and the real world.

Why Functions Matter

Functions are fundamental to:

• Describing relationships between quantities
• Making predictions based on patterns
• Understanding graphs
• All higher mathematics
• Computer programming
• Science and engineering

When students understand functions, they've grasped a concept they'll use throughout their mathematical journey.

The Big Idea: Input → Output

The Function Machine

    Input (x)
        ↓
    +-------+
    |       |
    |  f(x) |  ← The function (the rule)
    |       |
    +-------+
        ↓
    Output (y)

Example:

• Rule: "Double and add 1"
• Input: 5
• Output: 2(5) + 1 = 11

The Key Rule

Each input produces exactly ONE output.

Think of it this way:

• OK: Different inputs → same output (both 3 and -3 go to 9)
• OK: Each input → different output (1→2, 2→4, 3→6)
• NOT OK: Same input → different outputs (3→5 AND 3→7)

Function Notation

Reading f(x)

f(x) is read "f of x" and means "the output of function f when x is the input."

f(x) = 2x + 3

f(5) = 2(5) + 3 = 13

"f of 5 equals 13"

The notation does NOT mean f times x!

Evaluating Functions

Given f(x) = x² - 4x + 1, find:

f(3):

f(3) = (3)² - 4(3) + 1
     = 9 - 12 + 1
     = -2

f(-2):

f(-2) = (-2)² - 4(-2) + 1
      = 4 + 8 + 1
      = 13

f(a):

f(a) = a² - 4a + 1

f(x + 1):

f(x + 1) = (x + 1)² - 4(x + 1) + 1
         = x² + 2x + 1 - 4x - 4 + 1
         = x² - 2x - 2

Different Function Names

Functions can use any letter:

• f(x), g(x), h(x) are common
• P(t) might represent population over time
• A(r) might represent area as a function of radius

Representing Functions

1. As an Equation

f(x) = 3x - 5
y = x² + 2
g(n) = n/2 + 4

2. As a Table

x  |  f(x)
---------
1  |   4
2  |   7
3  |  10
4  |  13

Pattern: f(x) = 3x + 1

3. As Ordered Pairs

{(1, 4), (2, 7), (3, 10), (4, 13)}

Each pair is (input, output) or (x, f(x)).

4. As a Graph

    y
    |
 13 +           *
    |
 10 +        *
    |
  7 +     *
    |
  4 +  *
    |
    +--+--+--+--+-- x
       1  2  3  4

5. As a Verbal Description

"Triple the input and add one."

6. As a Mapping Diagram

Input    Output
  1   →    4
  2   →    7
  3   →    10
  4   →    13

Is It a Function? Testing Relations

The Definition Test

A relation is a function if each input has exactly ONE output.

Example 1: {(1, 3), (2, 5), (3, 7), (4, 9)}

• Each x-value appears once
• ✓ This IS a function

Example 2: {(1, 3), (2, 5), (1, 7), (4, 9)}

• The input 1 appears twice with different outputs (3 and 7)
• ✗ This is NOT a function

Example 3: {(1, 5), (2, 5), (3, 5), (4, 5)}

• Each input has one output (all happen to be 5)
• ✓ This IS a function

The Vertical Line Test (for Graphs)

If ANY vertical line crosses the graph more than once, it's NOT a function.

Function (passes test):

    y
    |     /
    |   /
    | /
    +--------x

   Any vertical line crosses once ✓

Not a function (fails test):

    y
    |   O
    |  / \
    | |   |
    |  \ /
    +---O----x

   A vertical line can cross twice ✗

Why the Test Works

A vertical line represents ONE x-value. If the line hits the graph twice, that means one x-value has two y-values—breaking the function rule.

Domain and Range

Definitions

Domain: All possible INPUT values (x-values)
Range: All possible OUTPUT values (y-values)

Think: "You DOMAIN the INputs, the outputs are in RANGE"

Finding Domain and Range from a Table

x  |  y
--------
1  |  4
2  |  7
3  |  4
5  |  10

Domain: {1, 2, 3, 5}
Range: {4, 7, 10} (note: 4 is only listed once)

Finding Domain and Range from a Graph

    y
  8 +     *
    |    *
  4 +   *
    |  *
  0 +--+--+--+--+-- x
       1  2  3  4

Domain: 1 ≤ x ≤ 4 (or {1, 2, 3, 4} if discrete)
Range: 0 ≤ y ≤ 8

Restrictions on Domain

Division: Can't divide by zero

f(x) = 1/(x-3)
Domain: All real numbers except x = 3

Square Roots: Can't take square root of negative (for real numbers)

f(x) = √(x-2)
Domain: x ≥ 2

Example: Finding Domain from Equation

f(x) = 5/(x + 4)

Ask: "What x-value would cause a problem?"

• Dividing by zero: x + 4 = 0 → x = -4

Domain: All real numbers except x = -4

Function Operations

Finding Output from Input

Given f(x) = 2x - 5, find f(7):

f(7) = 2(7) - 5 = 14 - 5 = 9

Finding Input from Output

Given f(x) = 2x - 5, find x when f(x) = 11:

11 = 2x - 5
16 = 2x
x = 8

Composing Functions

If f(x) = 2x and g(x) = x + 3:

f(g(x)) means "do g first, then f":

f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6

g(f(x)) means "do f first, then g":

g(f(x)) = g(2x) = 2x + 3

Notice: f(g(x)) ≠ g(f(x)) usually!

Real-World Functions

Everyday Examples

Temperature conversion:

F(C) = (9/5)C + 32

Taxi fare:

Cost(miles) = 2.50 + 1.75(miles)

Grading:

Grade(score) = ...
90-100 → A
80-89 → B
70-79 → C
...

Cell phone plan:

Bill(data) = 40 + 10(extra GB used)

Not Everything Is a Function!

Person → height: Function ✓ (one person, one height)
Height → person: NOT a function (many people can be the same height)

Year → president: Function ✓ (one president per year)
President → year: NOT a function (some served multiple years)

Hands-On Activities

Function Machine Game

• One student is the "machine" with a secret rule
• Others give inputs, machine gives outputs
• Players try to guess the rule

Input: 4   Output: 11
Input: 1   Output: 5
Input: 6   Output: 15

Rule: f(x) = 2x + 3

Mapping Diagram Creation

Create mapping diagrams for real relationships:

• Students → birth month
• States → number of letters in name
• Identify which are functions!

Vertical Line Test Practice

Give students various graphs. They use a ruler (vertical line) to test each one.

Function Scavenger Hunt

Find real-world examples of functions:

• Vending machines (selection → item)
• Calculators (input → output)
• Speed limits (location → speed)

Build-a-Function

Start with: "Everyone pick a number."
Apply operations: "Double it. Add 3. Square it."
Discuss: Is the result predictable from the input?

Common Mistakes and How to Fix Them

Mistake 1: Thinking f(x) Means f Times x

Wrong: f(3) = f × 3

Fix: f(3) means "plug 3 into the function."
If f(x) = x², then f(3) = 3² = 9.

Mistake 2: Confusing Domain and Range

Fix: Domain = inputs (x-values, left-right on graph)
Range = outputs (y-values, up-down on graph)

Mistake 3: Saying Same Output = Not a Function

Wrong: "(1, 5) and (2, 5) means it's not a function"

Fix: Same OUTPUT is fine! It's same INPUT with different outputs that breaks the rule. f(1) = 5 and f(2) = 5 is perfectly valid.

Mistake 4: Missing Domain Restrictions

Wrong: f(x) = √x has domain "all real numbers"

Fix: Can't take square root of negatives! Domain: x ≥ 0

Mistake 5: Evaluating f(x+2) as f(x) + 2

Wrong: If f(x) = x², then f(x+2) = x² + 2

Fix: Replace EVERY x with (x+2):
f(x+2) = (x+2)² = x² + 4x + 4

Practice Ideas for Home

Function Detective

Look for functions in daily life:

• What's the function that converts hours to minutes?
  f(h) = 60h
• What's the function that calculates the area of a square?
  A(s) = s²

Input-Output Tables

Create tables and find the rule:

x  |  y
--------
1  |  1
2  |  4
3  |  9
4  | 16

Rule: f(x) = x²

Domain Restrictions

List functions and identify domain restrictions:

• f(x) = x + 5 (all real numbers)
• f(x) = 1/x (all except 0)
• f(x) = √x (x ≥ 0)

Real-World Modeling

Write functions for situations:

• Cost of gas: C(g) = 3.50g
• Perimeter of a square: P(s) = 4s
• Distance traveled: d(t) = 60t

Connecting to Future Concepts

Linear Functions (Next Topic!)

When f(x) = mx + b, the function is LINEAR.

f(x) = 3x + 2 graphs as a straight line

Quadratic Functions

When f(x) = ax² + bx + c, the function is QUADRATIC.

f(x) = x² - 4 graphs as a parabola

Exponential Functions

When f(x) = a · bˣ, the function shows exponential growth/decay.

f(x) = 2ˣ doubles with each increase in x

Trigonometric Functions

sin(x), cos(x), tan(x) are functions of angles.

Calculus

Derivatives and integrals are operations ON functions:

• "What's the rate of change of this function?"
• "What's the area under this function?"

The Bottom Line

A function is simply a consistent rule: each input produces exactly one output. This simple idea is incredibly powerful.

Key points for students:

• One input → one output (the defining rule)
• f(x) notation names the function and shows the input
• Domain = possible inputs
• Range = possible outputs
• Vertical line test checks graphs

Functions describe relationships between quantities—how one thing depends on another. This concept underlies virtually all of higher mathematics, from algebra through calculus and beyond.

When students truly understand functions, they're not just learning a topic—they're gaining a lens through which to view mathematical relationships everywhere.